Question

identify the coordinates of any local and absolute extreme points and inflection points. graph the function.

$y = 12x^{3}-x^{4}=x^{3}(12 - x)$

identify the coordinates of the local maximum points. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

○ a. the local maximum point(s) is/are
(use a comma to separate answers as needed. type an ordered pair. do not use commas in the individual coordinates.)

○ b. there are no local maximum points.

Explanation:

Step1: Find the first derivative

Use the power rule \((x^n)^\prime=nx^{n - 1}\).
If \(y = 12x^{3}-x^{4}\), then \(y^\prime=36x^{2}-4x^{3}=4x^{2}(9 - x)\)

Step2: Find the critical points

Set \(y^\prime = 0\), so \(4x^{2}(9 - x)=0\)
Solving \(4x^{2}(9 - x)=0\) gives \(x = 0\) or \(x=9\)

Step3: Use the second - derivative test

Find the second derivative. \(y^{\prime\prime}=72x-12x^{2}=12x(6 - x)\)

• When \(x = 0\): \(y^{\prime\prime}(0)=12\times0\times(6 - 0)=0\), the second - derivative test is inconclusive.
• When \(x = 9\): \(y^{\prime\prime}(9)=12\times9\times(6 - 9)=12\times9\times(- 3)=-324<0\)

Since \(y^{\prime\prime}(9)<0\), the function has a local maximum at \(x = 9\)
Substitute \(x = 9\) into the original function \(y=12x^{3}-x^{4}\)
\(y=12\times9^{3}-9^{4}=12\times729 - 6561=8748-6561 = 2187\)

Answer:

A. The local maximum point(s) is/are \((9,2187)\)